let D1, D2 be non empty Subset-Family of Omega; ( ( for B being set holds
( B in D1 iff ( B in Sigma & B c= A & A \ B is thin of P ) ) ) & ( for B being set holds
( B in D2 iff ( B in Sigma & B c= A & A \ B is thin of P ) ) ) implies D1 = D2 )
assume that
A13:
for B being set holds
( B in D1 iff ( B in Sigma & B c= A & A \ B is thin of P ) )
and
A14:
for B being set holds
( B in D2 iff ( B in Sigma & B c= A & A \ B is thin of P ) )
; D1 = D2
for B being object holds
( B in D1 iff B in D2 )
hence
D1 = D2
by TARSKI:2; verum